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Abstract

Coupled mode theory for waveguide arrays is extended to next-nearest neighbor interactions using propagation equations. Both lateral diffraction and propagation of Floquet-Bloch waves are altered respectively by extra coupling and non-orthogonality between isolated waveguide modes. The analytical formula describing the distortions of the diffraction relation is validated by direct numerical simulation for weakly coupled InP and GaAs shallow ridge waveguides and for strongly coupled Si-SiO2 buried strip waveguides. The impact of extended coupled mode theory on propagation and diffraction design in waveguide arrays is discussed with reference to available experimental work.

Figures (10)

Variation of the effective indices of the isolated ridge waveguide nirw with ridge width Lr for the standard structure in InP (a) and GaAs (b) systems: fundamental mode (green symbols), mode 1 (orange), mode 2 (red), with horizontal polarization in full lines and vertical one in dashed lines. Planar waveguide levels nlpw and nupw are shown by dark green horizontal lines and their middle by light green ones. The supermode bands for C=1.3mm−1, Lr=3µm (InP) and C=0.3mm−1, Lr=4µm (GaAs) together with the levels of a 7-ridge array are illustrated by violet symbols. In (c) all curves are shown to coincide when drawn in reduced coordinates.

Model parameters as a function of ridge spacing S for horizontal polarization in InP-based 7-ridge structure with Lr=3µm. Calc, mod, fit in the captions indicate respectively the results of the calculation from the isolated mode, an exponential approximation of this mode, and the values giving the best fit of the FEM levels by the model diffraction relation.

X section of the fundamental mode of the 7-ridge array in the InP system (blue circles), with Lr=3µm and S=10µm (a) or S=4µm (b). The blue line is an adjustment by a combination of individual modes (violet lines) merely described by an exponential tail with weights given by the CMT model (red squares).

Variation of η, ξ, ζ, and C/Cmax with reduced ridge period s=S/Levan for all waveguide structures considered. Only values validated by FEM data are shown. Lines indicate the predictions of Eqs. (A10) and A13.

Schematic decomposition of the dielectric constant profile of shallow ridge waveguide arrays in vacuum in (X,Y) plane. Light blue and dark blue dashed vertical lines on the left hand side indicate interfaces where boundary conditions must be written respectively in “etched” and “ridge” planar waveguide regions.